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Theorem summodclem2 11408
Description: Lemma for summodc 11409. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
Hypotheses
Ref Expression
isummo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
isummo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
summodclem2.g 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
Assertion
Ref Expression
summodclem2 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2
Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5530 . . . . 5 (𝑚 = 𝑎 → (ℤ𝑚) = (ℤ𝑎))
21sseq2d 3200 . . . 4 (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑎)))
31raleqdv 2692 . . . 4 (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴))
4 seqeq1 10466 . . . . 5 (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
54breq1d 4028 . . . 4 (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
62, 3, 53anbi123d 1323 . . 3 (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
76cbvrexv 2719 . 2 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8 simplr3 1043 . . . . . . . . 9 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9 simplr1 1041 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ (ℤ𝑎))
10 uzssz 9565 . . . . . . . . . . . 12 (ℤ𝑎) ⊆ ℤ
119, 10sstrdi 3182 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ ℤ)
12 1zzd 9298 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 1 ∈ ℤ)
13 simprl 529 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℕ)
1413nnzd 9392 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℤ)
1512, 14fzfigd 10449 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ∈ Fin)
16 simprr 531 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
17 f1oeng 6775 . . . . . . . . . . . . . 14 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
1815, 16, 17syl2anc 411 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ≈ 𝐴)
1918ensymd 6801 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ≈ (1...𝑚))
20 enfii 6892 . . . . . . . . . . . 12 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
2115, 19, 20syl2anc 411 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ∈ Fin)
22 zfz1iso 10839 . . . . . . . . . . 11 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
2311, 21, 22syl2anc 411 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24 isummo.1 . . . . . . . . . . . . 13 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
25 simplll 533 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26 isummo.2 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2725, 26sylan 283 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
28 eleq1w 2250 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
2928dcbid 839 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
30 simpr2 1006 . . . . . . . . . . . . . . 15 (((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴)
3130ad2antrr 488 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴)
32 simpr 110 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → 𝑘 ∈ (ℤ𝑎))
3329, 31, 32rspcdva 2861 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → DECID 𝑘𝐴)
34 summodclem2.g . . . . . . . . . . . . 13 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
35 eqid 2189 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑔𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑔𝑛) / 𝑘𝐵, 0))
36 simprll 537 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37 simpllr 534 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38 simplr1 1041 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑎))
39 simprlr 538 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
40 simprr 531 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4124, 27, 33, 34, 35, 36, 37, 38, 39, 40summodclem2a 11407 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
4241expr 375 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
4342exlimdv 1830 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
4423, 43mpd 13 . . . . . . . . 9 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45 climuni 11319 . . . . . . . . 9 ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
468, 44, 45syl2anc 411 . . . . . . . 8 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
4746anassrs 400 . . . . . . 7 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48 eqeq2 2199 . . . . . . 7 (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦𝑥 = (seq1( + , 𝐺)‘𝑚)))
4947, 48syl5ibrcom 157 . . . . . 6 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
5049expimpd 363 . . . . 5 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
5150exlimdv 1830 . . . 4 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
5251rexlimdva 2607 . . 3 (((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
5352r19.29an 2632 . 2 ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
547, 53sylan2b 287 1 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 835  w3a 980   = wceq 1364  wex 1503  wcel 2160  wral 2468  wrex 2469  csb 3072  wss 3144  ifcif 3549   class class class wbr 4018  cmpt 4079  1-1-ontowf1o 5230  cfv 5231   Isom wiso 5232  (class class class)co 5891  cen 6756  Fincfn 6758  cc 7827  0cc0 7829  1c1 7830   + caddc 7832   < clt 8010  cle 8011  cn 8937  cz 9271  cuz 9546  ...cfz 10026  seqcseq 10463  chash 10773  cli 11304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-fz 10027  df-fzo 10161  df-seqfrec 10464  df-exp 10538  df-ihash 10774  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-clim 11305
This theorem is referenced by:  summodc  11409
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